Roman Zwicky (Southampton) , strong-bsm workshop Bad Honnef

Transcription

1 Hyperscaling rela.on for the conformal window Roman Zwicky (Southampton) , strong-bsm workshop Bad Honnef

2 Overview Intro: Motivation & introduction conformal window studies [4 slides] Part I: mass-deformed conformal gauge theories (observables) - hyperscaling laws of hadronic observables e.g. f[0 ++ ] ~ m η(γ * ) [8 slides] Part II: the quark condensate -- various appraoches [4 slides] lattice material (Del Debbio s talks) walking technicolour (Shrock, Sannino, others). Del Debbio & RZ PRD 10 & PLB 11

3 types of gauge theories Adjustable: gauge group SU(Nc) -- Nf (massless) fermions -- fermion irrep Focus on asymptotically free theories (not many representations) 2) well-defined on lattice 2) chance for unification in TC 11? 16 nf Nc =3 TC-models: QCD-type walking-type IR-conformal Susskind-Weinberg 79 Holdom 84 Dietrich-Sannino 04 Luty-Okui 04 confinement & chiral symmetry breaking SU L (n F ) x SU R (n F ) SU V (n F ) the latter is dynamical eletroweak symmetry breaking MW = g fπ (TC)

4 Conformal window (the picture) N=1 SUSY SU(N) non-susy N f N f 8 fund 15 fund A 4 2A 2 2S adj just below pert. BZ/BM fixed pt: lower line BZ/BM fixed pt electromagnetic dual assume in between conformal use βnsvz(γ*)= 0 to get γ* N c upper line AF weak coupling strong coupling 5 QCD 2S adj β 0 tuned small α s 2π = β 0 β 1 1 lower line Dyson-Schwinger eqs predict chiral symmetry breaking (lattice results later...) N c γ* strong =1 (unitarity bound QQ state) γ* strong 1 DS eqs ladder

5 canonical dimension d: anomalous dimension (classical) e.g. fermion field q: d q = D 1 composite: d qq = D 1 2 anomalous dimension γ: (quantum corrections) physical scheme-dependent γ = γ + γ 0 (g g )+O((g g ) 2 )... - significance: change of renormalization scale μ μ : scaling O(µ) =O(µ ) ( ) γ µ µ (1 + O(ln µ, /µ )) leading correction - QCD: UV-fixed point (asymptotic freedom) - γ * ( g * = 0)= 0 (trivial) - correction RGE (logarithmic) - our interest: IR fixed point non-trivial - γ * 0 (large?)

6 scaling dimension scaling dimension: Δ scaling = d canonical + γ anomalous L = i C i (µ)o i (µ) relevant 4 Δ marginal irrrelevant Value of Δ O is a dynamical problem (= d O at trivial fixed-point, g * =0) - unitarity bounds Δ Scalar 1 etc Mack 77 - Δ OO Δ O Δ O generally (except SUSY and large-n c ) gauge theory: expect qq to be most relevant operator Δ qq = 3 + γ qq = 3 - γ m (γ m = γ * at fixed point) γ * is a very important parameter for model building Ward-identity

7 Part I: Observables for Monte Carlo (lattice) for conformal gauge theories -- parametric control --

8 Part I: Observables in a CFT? or how to identify a CFT (on lattice) pure-cft: Vanishing β-function & correlators (form 2 & 3pt correlators known) e.g. <O(x)O(0)> ~ (x 2 ) -Δ deformed-cft: Lattice: quarks massive / finite volume consider mass-deformed conformal gauge theories (mcgt) * L = L CGT m qq * hardly related to 2D CFT mass deformation a part of algebra and therefore integrability is maintained

9 if mass-deformation relevant Δqq = 3 - γ* < 4 theory flows away from fixed-point (likely) physical picture: (Miransky 98) finite mq ; quarks decouple pure YM confines (string tension confirmed lattice) hadronic spectrum signature: hadronic observables (masses, decay constants) hypothesis: hadronic observables 0 as mq 0 (conformality restored) O m η O (1 +..), η O (γ ) > 0 If fct ηo known: a) way to measure γ* b) consistency test through many observable

10 Mass scaling from trace anomaly & Feynman-Hellman thm trace/scale anomaly: Adler et al, Collins et al N.Nielsen 77 Fujikawa 81 θα α on shell = 1 2 βg2 + N f m(1 + γ m ) qq β = 0 & H(p) H(k) =2E p δ (3) (p k) 2M 2 h = N f (1 + γ )m H qq H reminiscent GMOR-relation Feynman-Hellman thm: E λ λ = ψ(λ) Ĥ(λ) λ ψ(λ) idea: ψ(λ) ψ(λ) λ =0 applied to our case (λ m) : m M 2 H m = N f m H qq H combined with GMOR-like: m M H m = 1 1+γ M H M H m 1 1+γ scaling law for all masses

11 Brief comparison with QCD QCD-spectrum m ρ = O(Λ QCD ) + m q m B = m b + O(Λ QCD ) m π = O( (m q Λ QCD ) 1/2 ) mcgt-spectrum m ALL = m 1/(1+γ * ) O(Λ ETC γ * /(1+γ * ) ) Breaking global flavour symmetry : SU L (n F ) x SU R (n F ) SU V (n F ) (chiral symmetry) QCD: mcgt: CGT: spontaneous yes * no no explicit (mass term) yes yes no confinement yes yes no * Fπ 0 m 0 order parameter no chiral perturbation theory in mcgt (pion not singled out -- Weingarten-inequality still applies)

12 local matrix element: Hyperscaling laws from RG O 12 (g, ˆm m µ,µ) ϕ 2 O ϕ 1 physical states no anomalous dim. 1. O 12 (g, ˆm, µ) =b γ O O 12 (g, ˆm,µ ), RG -trafo O12 µ = bµ' g = b 0+γ g g ˆm = b 1+γ ˆm, y m = 1 + γ, γ g < 0 (irrelevant) 2. O 12 (ˆm,µ )=b (d O+d ϕ1 +d ϕ2 ) O 12 (ˆm,µ) change physical units 3. Choose b s.t. ˆm =1 trade b for m Hyperscaling relations master equation O 12 (ˆm, µ) (ˆm) ( O+d ϕ1 +d ϕ2 )/(1+γ ) * From Weinberg-like RNG eqs on correlation functions (widely used in critical phenomena)

13 Applications: master formula (local matrix element): ϕ 1 O ϕ 2 m ( O+d ϕ1 +d ϕ2 )/(1+γ ) 1. hadronic masses: 2M 2 h = N f (1 + γ )m H qq H m 2 1+γ alternative derivation vacuum condensates: G 2 m 4 1+γ, qq m 3 γ 1+γ more later on decay constants: φ = H(adronic) N.B. (Δ H = d H = -1 choice)

14 Remarks S-parameter: (q µ q ν q 2 g µν )δ ab Π V A (q 2 )=i S =4πΠ V A (0) [pion pole] d 4 xe iq x 0 T (V µ a (x)v ν b (0) (V A)) 0 hadronic representation: Π V A (q 2 ) f 2 V m 2 V q2 f 2 A m 2 A q2 m 2 P f 2 P q difficulties: a) non-local b) difference (density not positive definite) modulo (conspiracy) cancellations improve on nonperturbatve computations (lattice, FRG, DSE...) Π W TC V A (0) O(m 1 ) Π mcgt V A (0) O(m 0 ) pion-pole Π mcgt V A (q 2 ) m2/y m q 2 for q 2 (Λ ETC ) 2 Sannino 10 free theory

15 Summary - Transition low energy (hadronic) observables carry memory of scaling phase m H m 1 1+γ f H(0 ) m 2 γ 1+γ qq m 3 γ 1+γ all quantities scale with one parameter -- witness relations between the zoo critical exponents α, β, γ, ν.. = hyperscaling clarify: heavy quark phase and mcgt are parametrically from similar: m B(0 ) m b m H(0 ) m 1 1+γ distinct: f B(0 ) m 1/2 f H(0 ) m 2 γ 1+γ

16 Part 2: story quark condensate <qq> the most relevant operator important for conformal TC/ partially gauged TC models

17 How does CFT react to a perturbation Unparticle area I. couple CFT Higgs-sector: criteria breaking (NDA): L eff C O H 2 VEV C Ov 2 Λ 4 B Cv2 Λ O B Λ B (Cv 2 ) 1 4 O Fox, Rajaraman, Shirman 07 II. Heuristics: deconstruct the continuous spectrum of a 2-function. Infinite sum of adjusted particles can mimick continuous spectrum Stephanov O(x) n f n ϕ n (x); ϕ n O 0 f n, { 0.5 f n = δ 2 (Mn) Mn 2 = nδ 2 tadpole & mass term as potential find new minimum V eff = m n f nϕ n 1/2 n M 2 nϕ 2 n. Delgado, Espinoso, Quiros 07

18 minimise - solve - reinsert: δ ϕn V eff =0 mf n + M 2 nϕ n =0 ϕ n = mf n /M 2 n O n f n ϕ n m n f 2 n M 2 n δ 0 m Λ 2 UV Λ 2 IR s O 3 ds result depends on IR and UV physics need model(s) III. Within conformal gauge theory generic operator: O qq within gauge theories N.B. 4D only known CFT gauge theories. why? Sannino RZ 08 - Λ UV Λ ETC where Δ qq 3 (Λ ETC ) 2 -effect - Λ IR : (m constituent ) Δqq ~ <qq> generalising Politzer OPE Solve self-consistency condensate eqn 1 A/q B = 1 OPE extension /q 4g + 2 q 4 of pole mass m = B A - 10: Λ IR : = m H m 1/(1+γ * ) O(Λ ETC γ * /(1+γ * ) ) agrees depending on value γ* Del Debbio, RZ Sep 10

19 IV. Generalized Banks-Casher relation Banks & Casher 80 à la Leutwyler & Smilga 92 : Green s function: q(x) q(y) = n u n (x)u n (y) m iλ n, where /Du n = λ n u n qq V = dx V q(x)q(x) λ n λ = n 2m V = 2m µf dλ ρ(λ) 0 m 2 + λ }{{ 2 } IR part λ n >0 2m 5 dλ µ F λ 4 1 m 2 + λ 2 n ρ(λ) m 2 + λ 2 } {{ } twice subtracted V 2m 0 dλρ(λ) m 2 + λ 2 + γ }{{} 1 m + γ 2 m 3 }{{} Λ 2 ln Λ UV UV IR-part: change of variable: - QCD: η qq =0 ρ(0) = π qq - mcgt: another way to measure γ * ρ(λ) λ 0 λ η qq qq m 0 UV-part: known from perturbation theory (scheme dependent) m η qq Banks, Casher 80 DeGrand 09 DelDebbio RZ 10 May

20 Summary I. NDA Λ IR ~ m 1/(1+γ * ) ok II. Deconstruction: O Λ 2 UV Λ 2 IR s O 3 ds model dependence III. Model mcgt & deconstruction - Λ UV properly identified thanks asymptotic freedom - Λ IR m constituent generalized QCD (Polizter OPE) ok dep value γ * IV. Model mcgt & generalized Banks-Casher - clean seperation of IR & UV everything consistent e.g. can use m H ~ m 1/(1+γ * ) in deconstruction as well

21 Epilogue Identified universal hyperscaling laws in mass deformation valid for any conformal theory in the vicinity of the fixed point (small mass) One thought: 1) CGT likely phase diagram as compared to walking theory 2) CGT instable m-deformation. Any quark that receives mass can be expected to decouple and finally drive the theory into a confining phase and the remaining quarks can undergo chiral symmetry breaking and thus dynamical electroweak symmetry breaking. Contrasts: Dynamical stability of local gauge symmetry... Forster, Nielsen & Ninomiya 80 Danke für die Aufmerskamkeit!

22 Backup slides...

23 Some relevant/useful references Miransky hep-ph/ spectrum (with mass) as signal of conformal window works with pole mass -- weak coupling regime ΛYM m Exp[-1/bYM α * ] glueballs lighter than mesons Luty Okui JHEP 96 conformal technicolor propose spectrum as signal of cw Dietrich/Sannino PRD 07 conformal window SU(N) higher representation using Dyson-Schwinger techniques known from WTC Sannino/RZ PRD 08 <qq> done heuristically IR and UV effects understood 0905 DelDebbio et al ArXiv 0907 Mass lowest state from RGE equation DeGrand scaling <qq> stated ArXiv 0910 DelDebbio RZ ArXiv 0905 scaling of vacuum condensates, all lowest lying states DelDebbio RZ ArXiv 0909 scaling extended to entire spectrum and all local matrix elements

24 Del Debbio, RZ Sep 10 Mass & decay constant trajectory At large-n c neglect width g Hn 0 O H n (decay constant) (q 2 ) x eix q 0 O(x)O(0) 0 = n g H n 2 q 2 +M 2 Hn In limit m 0 (scale invariant correlator) (q 2 )= 0 ds s 1 γ q 2 +s +s.t (q 2 ) 1 γ Solution are given by: MH 2 n α n m 2 1+γ, gh 2 n α n(α n ) 1 γ m 2(2 γ ) 1+γ where α n arbitrary function (corresponds freedom change of variables in ) QCD expect α n ~ n (linear radial Regge-trajectory) (few more words) For those who know: resembles deconstruction Stephanov 07 difference physical interpretation of spacing due to scaling spectrum

25 Addendum (bounds scaling dimension) assume add L = mqq (N.B. not a scalar under global flavour symmetry!) using bootstrap ( associative OPE on 4pt function) possible to obtain upper-bound on scaling dimension Δ of lowest operator in OPE non-singlet Rattazzi, Rychkov Tonni & Vichi 08 singlet Δ 4 allows for Δ qq to be: Rattazzi, Rychkov & Vichi 10 very close to unitarity bound! 1.35 still rather close to unitarity bound good news for Luty s conformal TC

26 QCD observable Gell-Mann Oakes Renner: mcgt not so directly observable very important for Technicolour f 2 πm 2 π = 2m qq L eff = f ermion masses FCNC